A three-dimensional theoretical model for estimating the thermal residual stresses in micro multilayer ceramic capacitors Wu-Gui Jiang a , Xi-Qiao Feng a, * , Cewen Nan b a Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China b State Key Laboratory of New Ceramics and Fine Processing and Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, China Received 20 April 2007; received in revised form 29 July 2007; accepted 13 September 2007 Available online 20 September 2007 Abstract A three-dimensional theoretical model is proposed to estimate the thermal residual stresses in multilayer ceramic capacitors (MLCCs) based on the effective medium method in micromechanics of composites. The MLCC is first decomposed into some sub-blocks with dif- ferent material properties, and then the classical lamination theory and the combined Voigt’s and Reuss’ assumptions are adopted to calculate the residual stress distributions. The results obtained from this theoretical model show a good agreement with the numerical results of finite element method. The effects of such key factors as the dielectric layer thickness, the number of dielectric layers, the cover layer sizes, as well as the length and width creepage paths on the distributions of residual stresses are all examined. The suggested the- oretical method provides a convenient tool for estimating the residual stresses in MLCCs as a function of characteristic sizes and man- ufacturing process parameters and, therefore, is helpful for the optimal design of MLCCs with enhanced performance. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: A. Layered structures; A. Multilayer ceramic capacitor; B. Modeling; C. Residual stress; C. Laminate theory 1. Introduction As an important type of surface-mount devices and information materials, multilayer ceramic capacitors (MLCCs) have been widely used in a broad range of appli- cations from household devices to satellites because of their advantages of high capacitance density and small device sizes [1,2]. An MLCC has the so-called ‘‘brick’’ structure consisting of dielectric layers (e.g. BaTiO 3 ) and internal metallic electrodes (e.g. Ni), which are laminated alter- nately with each other and co-fired at a high temperature up to about 1250 °C. The high-temperature sintering is a key step in fabrication process of MLCCs. After sintering, the component is cooled down to room temperature, which unavoidably induces residual stresses in MLCCs due to the mismatch of the thermal expansion coefficients between these two dissimilar materials. The residual stresses affect significantly the mechanical integrity of the devices and may cause damage and failure of the systems [3–5]. Some recent work has been directed towards investiga- tion on the residual stress states in MLCCs [6–13]. For example, Nakano et al. [6] measured the residual stress in MLCCs by X-ray diffraction (XRD) analysis and found that the stress increases as the number of the dielectric cera- mic layers increases. Saito et al. [7] investigated the uniaxial compressive stress and electric field responses of X5R-type Ni-MLCCs. Their results showed that the capacitance changes of the MLCCs exhibit a considerable dependence on the stresses. As the number of dielectric layers increases up to 500 and the dielectric layer thickness decreases to sev- eral microns, however, it becomes difficult to experimen- tally measure the detailed residual stress state in MLCCs. Recently, finite element methods (FEMs) have been used to numerically analyze the stress states in MLCCs [8–13]. Franken et al. [8] and Prume et al. [9] presented a 0266-3538/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2007.09.006 * Corresponding author. Tel.: +86 10 62772934; fax: +86 10 62781824. E-mail address: fengxq@tsinghua.edu.cn (X.-Q. Feng). www.elsevier.com/locate/compscitech Available online at www.sciencedirect.com Composites Science and Technology 68 (2008) 692–698 COMPOSITES SCIENCE AND TECHNOLOGY